Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a

Jewish-German The history of the Jews in Germany goes back at least to the year 321, and continued through the Early Middle Ages (5th to 10th centuries CE) and High Middle Ages (''circa'' 1000–1299 CE) when Jewish immigrants founded the Ashkenazi Jews, As ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who contributed important research in the field of

linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ ...

s. He was born in Moschin (Mosina) (located in

Grand Duchy of Posen The Grand Duchy of Posen (german: Großherzogtum Posen; pl, Wielkie Księstwo Poznańskie) was part of the Kingdom of Prussia, created from territories annexed by Prussia after the Partitions of Poland, and formally established following th ...

) and died in

Berlin Berlin is Capital of Germany, the capital and largest city of Germany, both by area and List of cities in Germany by population, by population. Its more than 3.85 million inhabitants make it the European Union's List of cities in the European U ...

Germany Germany, officially the Federal Republic of Germany (FRG),, is a country in Central Europe. It is the most populous member state of the European Union. Germany lies between the Baltic and North Sea to the north and the Alps to the sou ...

. He was buried in

Schöneberg Schöneberg () is a locality of Berlin, Germany. Until Berlin's 2001 administrative reform it was a separate borough including the locality of Friedenau. Together with the former borough of Tempelhof it is now part of the new borough of Te ...

in the St. Matthew's Cemetery. His grave in section H is preserved and listed as a grave of honour of the State of Berlin. He is the eponym of

Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations ...

s and functions, and the

Picard–Fuchs equation In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves. Definition Let :j=\frac be the j-invariant with g_2 an ...

. A

singular point Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...

''a'' of a

y''+p(x)y'+q(x)y=0

is called Fuchsian if ''p'' and ''q'' are

meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...

around the point ''a'', and have poles of orders at most 1 and 2, respectively. According to a theorem of Fuchs, this condition is necessary and sufficient for the regularity of the singular point, that is, to ensure the existence of two linearly independent solutions of the form :

y_j=\sum_^\infty a_(x-x_0)^,\quad a_0\ne0\,\quad j=1,2.

where the exponents

\sigma_j

can be determined from the equation. In the case when

\sigma_1-\sigma_2

is an integer this formula has to be

modified Modified may refer to: * ''Modified'' (album), the second full-length album by Save Ferris * Modified racing, or "Modifieds", an American automobile racing genre See also * Modification (disambiguation) * Modifier (disambiguation) Modifier may ...

. Another well-known result of Fuchs is the ''Fuchs's conditions'', the necessary and sufficient conditions for the non-linear differential equation of the form :

F\left(\frac,y,z\right)=0

to be free of movable singularities. An interest remark about him as a teacher during the period of his work at the

Heidelberg University } Heidelberg University, officially the Ruprecht Karl University of Heidelberg, (german: Ruprecht-Karls-Universität Heidelberg; la, Universitas Ruperto Carola Heidelbergensis) is a public research university in Heidelberg, Baden-Württemberg, ...

pertains to his manner of lecturing: his knowledge of the mathematics he was assigned to teach was so deep that he was wont not to prepare before giving a lecture — he would simply improvise on the spot, while exposing the students to the train of thought taken by mathematicians of the finest degree. Lazarus Fuchs was the father of Richard Fuchs, a German mathematician.

Selected works

* ''Über Funktionen zweier Variabeln, welche durch Umkehrung der Integrale zweier gegebener Funktionen entstehen'', Göttingen 1881. * ''Zur Theorie der linearen Differentialgleichungen'', Berlin 1901. * ''Gesammelte Werke'', Hrsg. von Richard Fuchs und Ludwig Schlesinger. 3 Bde. Berlin 1904–1909.

References

External links

* * *

G. B. Mathews 250px George Ballard Mathews, FRS (23 February 1861 – 19 March 1922) was an English mathematician. He was born in London. He studied at the Ludlow Grammar School which had instruction in Hebrew and Sanscrit as well as in Greek and Latin. He ...

(1902
Lazarus Fuchs

Nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...

66:156,7 (#1702). * * 1833 births 1902 deaths 19th-century German Jews Converts to Lutheranism from Judaism People from Poznań County People from the Grand Duchy of Posen Humboldt University of Berlin alumni University of Greifswald faculty 19th-century German mathematicians 20th-century German mathematicians 19th-century Lutherans Members of the Göttingen Academy of Sciences and Humanities Members of the Royal Society of Sciences in Uppsala {{Germany-mathematician-stub